Difference between revisions of "Incenter"
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| − | [[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with incenter ''I'', with angle bisectors (red), incircle (blue), and | + | [[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with incenter ''I'', with [[angle bisectors]] (red), [[incircle]] (blue), and [[incircle]] (green)]] |
The '''incenter''' of a [[triangle]] is the intersection of its (interior) [[angle bisector]]s. The incenter is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter. | The '''incenter''' of a [[triangle]] is the intersection of its (interior) [[angle bisector]]s. The incenter is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter. | ||
Revision as of 20:31, 15 September 2007
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The incenter of a triangle is the intersection of its (interior) angle bisectors. The incenter is the center of the incircle. Every nondegenerate triangle has a unique incenter.
Proof of Existence
Consider a triangle 
. Let 
be the intersection of the respective interior angle bisectors of the angles 
and 
. We observe that since lies on an angle bisector of , is equidistant from 
and 
; likewise, it is equidistant from 
and ; hence it is equidistant from and and and therefore lies on an angle bisector of 
. Since it lies within the triangle , this is the interior angle bisector of . Since is equidistant from all three sides of the triangle, it is the incenter.
It should be noted that this proof parallels that for the existance of the circumcenter.
The proofs of existance for the excenters is the same, except that certain angle bisectors are exterior.
Properties of the Incenter
The incenter of any triangle lies within the orthocentroidal circle.
The unnormalised areal coordinates of the incenter are 
